1. Introduction

1.1. Sandwich vs. OpenCell

Sandwich structures are well known in the field of structural mechanics. They are composed of two

thin plates (usually made from metal) separated with a core material (which can be made from

several materials, namely foam, wood, recycled paper, etc.). The main advantages of this type of

structures are the high specific strength and stiffness, the great variety of material combinations

and their low production cost.

But these structures have some disadvantages, namely the high cost of the recycling process.

When recycling these types of structures the thin plates have to be separated from the core material.

This can be a very expensive and troublesome process.

The company Ply Engenharia developed a

mechanical structure that presents the main advantage of a sandwich panel but doesn’t require

the existence of a different core material. The core is obtained from one of the external plates, by

cutting and bending a part of it, and then welding the fin to the opposite plate. Only one of the plates

is cut, so that the other one can withstand pressure

1

Optimization of reinforcements in panels considering plasticity

Segismundo de BragançaInstituto Superior Técnico

15/05/2012

Abstract:

The purpose of this work is the optimization of an OpenCell type panel. This kind of plate is presented as an alternative to the well known sandwich panels. It consists in a regular repetition of a hexagonal shaped cell.

Two sets of optimization processes are performed: the first, a linear static analysis of a plate and subsequent

comparison to a plate composed of cells with different geometry; the second, a non-linear analysis of a plate when subject to a load that caused plastic deformation.

The optimization process is made through the interface between Ansys, a FEA based software, and Matlab,

which has the optimization algorithms used. A computational model that would provide this interface is also developed.

The main focus of this work is the influence of the design variables in the plate behavior, and to study how

their optimal values change when going from the elastic to the plastic regime.

During the course of this work several parameters and indicators are used, in order to study and analyze the behavior of the plate in different situations, so that the scope of the obtained results isn’t limited to a specific

working situation of the plate, thereby simulating different types of applications.

Keywords: OpenCell, Structural optimization, Plasticity, Sandwich panels, Matlab, Ansys

l oads , i f necessary. Some o f the ma in configurations of an OpenCell panel can be seen

below:

It is obvious that the main disadvantage pointed out to the sandwich structures doesn’t exist here,

because the entire panel is made out of the same material.

1.2. Motivations and objectives

This is a recent innovation, so the available

literature isn’t as extensive as the existing for sandwich panels. Extensive tests have to be made

to understand the behavior of OpenCell panels.

This work consists on a structural optimization process to find an optimal cell geometry. The tests

to be performed can be divided in two main sections:

. A linear analysis of a plate in the elastic

regime and comparison to a previous work [1];

. A non-linear analysis of a plate in the plastic regime, considering two different objective

functions;

It’s also an objective of this work to obtain a computational model that allows us to analyze,

study and optimize an OpenCell type panel, and that makes the interface between the two softwares

to be used.

1.3. Expected results

Figure 1: OpenCell panels configuration

With this work we expect to obtain an optimal cell

configuration for both elastic and plastic regime. It’s also expected to obtain some insights on the

influence of a new design variable on the plate behavior. It’s also expected that the computational

model developed allows us to perform optimization analysis of the plate when subject to different types

of loads and boundary conditions.

1.4. Structure of the paper

The paper is structured as follows: section 2 is a brief review of the theoretical background used in

this paper. Section 3 describes the cell geometry, states the objective function, the design variables

and the constraints adopted. The overall configuration and properties of the plate are also

stated here. Section 4 is a brief introduction to the computational tools used to perform the structural

optimization and gives an overall idea of the optimization process. In section 5 the obtained

results are discussed and analyzed. Finally, section 6 is the conclusion of the work developed, and

some observations to future work are also made.

Section 2: Theoretical Review

2.1. Plate Theory

The definition of a plate is a solid bounded by two parallel planes which has two dimensions

substantially higher than the third one. There are several plate theories available, in which it is

important to mention the Classical Plate Theory developed by Kirchhoff and Love [2]. The

assumptions made in this theory make it appropriate to the analysis of thin plates [3].

The work of Reissner-Mindlin takes into account

the influence of the shear deformation along the thickness [4], making it appropriate to the analysis

of thick plates. This is the theory used by the FEA

2

software to perform the linear and non-linear analysis of the plate.

2.2. Optimization Algorithms

Two different optimization algorithms are used to

perform the structural optimization. The first is a genetic algorithm, which is part of the evolutionary

computing, an area of the artificial intelligence field. There are two seminal papers in the field: the work

of I. Rechenberg [5] and the work of J. Holland [6]. This type of algorithm tries to mimic the crossover

and the mutation of the DNA, and apply it to an optimization process. Unlike other algorithms, this

one consists on the analysis of different sets of solutions (population), choosing the solutions

(individuals) with the best fitness to make a new generation. The process is repeated until the best

solution is achieved.

The second algorithm is a gradient based one, belonging to a smaller group called Sequential

Quadratic Programming. These type of algorithms are based on the Karush-Kuhn-Tucker conditions

[7], which are necessary and sufficient conditions to an optimal solution. Gradient-based algorithms

need first-order information of the objective function in hands, which is then used to generate an

approximation of the Hessian matrix. In SQP algorithms this approximation is obtained through

the gradients of the Lagrange equations of the problem.

2.3. Plasticity

The elastic deformation is caused by the stretching

of interatomic bonds, which return to their original configuration when the load ceases. Plastic

deformation is caused by crystal slip, and ‘requires the breaking and re-forming of interatomic bonds

and the motion of one plane of atoms relative to another’ [8].

There are several yield criterion available in the literature, and in this work the von Mises criteria

was adopted [9]. It has the following general form:

Where α relates to the material hardening. If F(σ,α) is minor than zero, we are still in the elastic regime.

If F(σ,α) equals zero, plasticity occurs.

The second part of the equation is the hardening component. In this work the isotropic hardening

model is adopted. It can be translated by a uniform expansion of the yield surface in the stress space.

The main disadvantage of this model is that it doesn’t take into account the existence of the

Bauschinger effect [10], but since the load applied in the tests performed in this work is not reversed,

this effect is not important.

3: Cell Configuration

3.1. Cell geometry and design variables

In this paper a hexagonal cell is used, and two of its design variables can be seen below:

In the picture above b is the hole dimension and θ is the inclination of the fins. The fins are welded to

the lower plate. These are triangular shaped, which means that, theoretically, they end in a point. This

is not physically possible, so the part of the fin welded to the plate is a line. The length of this line

is directly related to the project variable b:

F(σ ,α ) = f (σ )−σ Y (α )

F(σ ,α ) = 12[(σ 1 −σ 2 )

2 + (σ 2 −σ 3)2 + (σ 3 −σ 1)

2 ]1/2 −σ Y (α )

Figure 2: Project variables

3

(1)

The third project variable is this numerical constant α, which directly influences the length of the

welding line. The image above shows that the height of the panel directly depends on the three

project variables. In the present paper two types of constraints are used. The first type consists of

inequality linear constraints:

Between brackets we have the values used. These constraints force the project variables to adopt

values between two limits, which are the minimum and maximum allowable, so that the cell doesn’t

loose its geometrical properties. The second type is a non-linear inequality constraint:

Figure 3: Geometric relations

binf (0.01) ≤ b ≤ (0.09)bsupθ inf (40) ≤θ ≤ (150)θsupα inf (0.15) ≤α ≤ (0.9)α sup

Where L is the distance between the hole and the fin edge, and F is the distance between the hole

and the cell edge, and both can be seen in figure 3. The inequality constraint relates all the project

variables, and it’s important when all the variables are used simultaneously. In this case it may

happen that the values of the project variables are between the linear constraints but, still, the fins

edges are outside the cell limits. In the above constraint, γ is a security parameter that sets the

gap between the hole and the edge of the cell.

3.2: Objective function

The objective function for the linear analysis is the minimization of the total energy of the plate, which

is the work of the external forces.

This is the same as the maximization of the plate stiffness.

For the non-linear analysis there are two objective

functions. The first is the minimization of the plastic equivalent strain, given by [11}:

This value is obtained for the element nodes. The objective function using this value is:

L ≤ γ F

F = a2− bsin60º

L = hcosθ = tan60º2

b(1−α )cosθ

cosθ tan60ºb(1−α )a2− bsin60º

≤ 2γ

[K]{d} = {F}→ {d}T [K]{d} = {d}T {F}min f (x) = {d}T {F}

Δε̂ pl =23

Δε pl{ }T [M ] Δε pl{ }⎛⎝⎜

⎞⎠⎟

Δε̂ pl dAΩ∫All load

increments

∑

4

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(2)

(3)

(4)

The second objective function is the minimization of the plastic work [11]:

Again, this value is obtained for the element nodes. The objective function using this value is:

In both the equations [M] is a constant matrix used by Ansys and Ω is the plate total area.

3.3: Plate

The complete plate consists on a regular repetition

of each cell described above. Each cell has a length of 0.2 m (a), and the plate has 5 x 5 cells.

The properties of the plate are in table 1.

a (m) 0.2

N 5 x 5

LT (m) 1 x 1

E (GPa) 70

Et (GPa) 7

σy (MPa) 100

υ 0.5

Δk = {σ}[M ]{depl}

Δk dAΩ∫

All load increments

∑

Figure 4: OpenCell plate

Table 1: Plate properties

3.4. Loads and boundary conditions

In all the optimization tests a distributed pressure on the bottom plate is used. The value of this

pressure changes from case to case, depending on the boundary conditions.

For the first part of the work we use only one type

of boundary condition: the four sides of the plate clamped (C-C-C-C). In the second part two

opposite sides are clamped and the other two sides free (C-C-F-F); and, finally, one side clamped, two

opposite sides free, and the other side simply supported (C-F-F-S).

4. Optimization Process

4.1: Ansys

The commercial software Ansys is a FEA software

which uses the Mindlin-Reissner theory for the analysis of the plate in question. In the first part of

the work this software is used to obtain the value of the total energy of the plate through a static linear

analysis. In the second part it’s used to obtain the total plastic work in equation (10), which is the work

made by the external forces during the plastic deformation, and the equivalent plastic strain,

calculated using equation (8).

This parameter contemplates all the values of the plastic strain vector, and can be seen as an

indicator of the overall plasticity on the plate. In fact, for a material that verifies the von Mises yield

criteria, we have:

Where λ is the plastic multiplier, a measurement of the overall plasticity.

The element chosen for this work was Shell 181,

‘suitable for analyzing thin to moderately thick shell structures’ [11], with eight nodes and six degrees of

Δε̂ pl = λ

5

(13)

(14)

(12)

freedom. This element is also suitable for analyzing plastically deformed structures.

4.2. Matlab

‘Matlab is a programming environment for algorithm

development, data analysis, visualization, and numerical computation’ [12]. This software has an

optimization toolbox with several available algorithms. Two different algorithms are used: a

genetic algorithm solver, based on the theory mentioned in section 2.2, and fmincon, which is a

SQP algorithm.

One difference between these two algorithms is that fmincon requires user given starting search

points, while genetic algorithms generate pseudo-random starting search points. So, in each

optimization process we start with the genetic algorithm solver, and use the optimal points

obtained as the starting points for fmincon.

We can see below a flowchart that describes the overall optimization process:

Figure 5: Process flowchart

5. Results and Discussion

5.1.1. Linear analysis (θ, b)

The first part of the work is a linear analysis of the plate and comparison with a similar optimization

made for a different geometry. In the first test the non-linear constraint isn’t used. This test is made

with a pressure of 5.000 N, and only two project variables are used. Since in the other work the

variable α didn’t exist, it couldn’t be used in this comparison. So that the maximum height of the two

panels can be equal its value is 0.1443. The results obtained with the other cell configuration are shown

in the first line:

Alg.Elastic Energy

[J]bopt [m]

Ɵopt [º] αopt

Fun. Evaluat.

fmincon 0.0840 0.16 76.8 44

Genético 0.0785 0.089 71.7 630

fmincon 0.0756 0.090 75.9 165

Both configurations exhibit a fin inclination different from 90º. The maximum height would be obtained

with a value of θ of 90º, which, as can be seen, is not the case. This inclination, despite lowering the

panel, increases its stiffness, reducing the maximum deformation.

The value of b is, as expected, the top constraint. It

maximizes the height of the panel and makes the plate more difficult to deform, because the cells

holes occupy the majority of the plate. The panel with the hexagonal cell has a lower objective

function. This happens because this cell has six fins (against the four fins of the quadrangular cells)

which makes the plate more difficult to deform. So its stiffness is higher. In the image below we can

see the optimal configuration for both cells:

Table 2: Case 1.1 results

6

5.1.2. Linear analysis (α, θ, b)

In this test the three project variables are used

simultaneously, and the optimization process is performed with and without the non-linear

constraint. The first two lines are the results using the non-linear constraint, and the last two without it:

Alg.Elastic Energy

[J]bopt [m]

Ɵopt [º] αopt

Fun. Evaluat.

Genetic 0.0710 0.089 72.92 0.245 6520

fmincon 0.0712 0.089 71.12 0.250 129

Genetic 0.06879 0.089 69.65 0.238 1050

fmincon 0.06773 0.090 72.9 0.238 165

The value of the project variable b is, as in the previous case, the maximum possible, maximizing

the hole. The objective function is lower because the three variables are used together, and so the

panel has more flexibility to find an optimal configuration.

The value of the additional project variable α shows

that the connection length between the fins and the plate is, approximately, one quarter of the value of

Figure 6: (i) Hexagonal cell optimal configuration (ii) Quadrangular cell optimal configuration

Table 3: Case 1.2 results

b. Also, as in the previous case, the fins have an inclination different from 90º.

The number of function evaluations with the non-

linear constraint is six times higher than without it, which implies a considerably higher computational

effort.

Even when the non-linear constraint isn’t used only the genetic algorithm optimal solution is not

acceptable. But when fmincon is used, the result is inside the solution space. The objective function

obtained with the non-linear constraint is higher than without it, which means this constraint makes

the optimization process convergence more difficult.

5.2. Non-Linear Analysis

The pressure applied in this case doesn’t cause

total plasticity of the plate because this could alter the quality of the numerical results. A uniform

pressure of 4 MPa is used.

5.2.1. Equivalent Plastic Strain (α, θ, b)

The optimization process is performed with and without the non-linear inequality constraint, and the

first two lines are the results with this constraint:

Alg. Plastic Strain

bopt [m]

Ɵopt [º] αopt

Fun. Evaluat.

Genetic 4.45 0.089 75.58 0.321 7650

fmincon 4.44 0.089 75.53 0.321 152

Genetic 4.50 0.089 81.04 0.299 1020

fmincon 4.39 0.089 85.10 0.298 303

The project variable b has the maximum allowable value. Like in the previous cases, this result

maximizes the space occupied by the cell hole. The fins inclination is higher than 5.1.2, which could

mean it’s loosing some stiffness. This is balanced

Table 4: Case 2.1 results

7

(i)

(ii)

by α, whose value is higher. This means the welding length is bigger, and so the height is lower.

Between 5.1.2 and this case there is only a difference of 1% in the height, despite that in this

situation the plate is subject to a higher pressure. The difference between a non-optimal and an

optimal solution can be seen in figure 7.

The zone in red means it’s undergoing plastic deformat ion. The plate wi th the opt imal

configuration presents a significantly lower plastic area. In the case of the optimal configuration the

plastic deformation takes place outside the holes (the hexagons in black lines), because it’s more

difficult to deform them. This is why, in the optimization process, the variable b tends to go to

its upper limit, so it maximizes the cell hole

The last two results, obtained without the non-linear constraint, don’t violate it. This means that both

algorithms naturally converge to a solution inside the solution space defined by the constraint. As in

case 5.1.2, the non-linear constraint causes a high number of function evaluations in the genetic

algorithm. This causes a very slow convergence rate. If we add this to the fact that the optimal

solution naturally verifies the non-linear constraint it’s reasonable to assume that its use is not

essential, only a verification of the obtained results.

5.2.2. Plastic work (α, θ, b)

In this case the objective function is the plastic work, which is given by equation (10), and the non-

linear constraint isn’t used.

Figure 7: SRAT (i) Non-optimal configuration (ii) Optimal configuration

Alg. Plastic Work [J]

bopt [m]

Ɵopt [º] αopt

Fun. Evaluat.

Genetic 5.995E8 0.089 81.37 0.394 2060

fmincon 5.009E8 0.089 84.00 0.296 389

The results are similar to the previous cases: b goes to its upper limit, maximizing the area of the

panel occupied by the hole, and consequently its stiffness; the fin inclination θ, as expected, presents

an inclination different from 90º, reducing the plate height but enhancing its stiffness; the variable α

has a similar value of the optimal results in case 5.2.1, being the connection length between the fin

and the lower plate approximately one third of b. All these value combined result in the same height of

case 5.2.1. The optimal configuration also verifies the non-linear constraint.

When using this objective function the convergence

is more difficult. The reason for this is because the value of the plastic work is 10E8 times higher than

the plastic strain, presenting a higher variation, causing a lower convergence speed. In fact, the

algorithm didn’t converge, it exceeded the generations limit, as can be seen below:

5.3. Boundary Conditions – C-C-F-F

Table 5: Case 2.2 results

Figure 8: fmincon convergence

8

(i) (ii)

In this case the plate has two opposite sides clamped and the other two free. The applied

pressure is 3.5 MPa.

5.3.1. Equivalent Plastic Strain and Plastic Work (α, θ, b)

In both cases the non-linear constraint isn’t taken

into account. The results obtained with both objective functions are in the table below, where

the first two lines are the equivalent plastic strain:

Alg. Object. Fun.

bopt [m]

Ɵopt [º] αopt

Fun. Evaluat.

Genetic 36.89 0.089 76.43 0.313 1220

fmincon 36.70 0.089 75.53 0.321 165

Genetic 5.48E9 J 0.077 58.83 0.268 2740

fmincon 2.77E9 J 0.089 75.96 0.366 215

The results are similar to the previous ones. The slight difference of results is in the inclination of the

fins θ. Its value is lower in the present case, lowering the panel height. This result can be

explained with the fact that, the plate having two free sides, its natural stiffness and resistance to

deformation are somewhat affected. The stiffness lost with these boundary conditions is compensated

with θ. The height of the panel in this case is, approximately, 10% lower than in 5.2.2.

As expected, all the results are inside the solution

space defined by the non-linear constraint.

5.4. Boundary Conditions – C-S-F-F

In cases 5.1, 5.2 and 5.3 the boundary conditions are symmetric in both x and y. This makes the

optimal solutions similar. In this case, there’s only symmetry in x, where the boundaries are free. In y

one boundary is clamped and the other simply supported. In this case only the equivalent plastic

strain is used as objective function.

Table 6: Case 3.1 results

5.4.1. Equivalent Plastic Strain (α, θ, b)

The results are shown below:

Alg. Plastic Strain

bopt [m]

Ɵopt [º] αopt

Fun. Evaluat.

Genetic 22.83 0.081 58.96 0.268 1040

fmincon 22.80 0.080 59.04 0.267 178

The results obtained here are different from the previous cases. The project variable b doesn’t have

its maximum value and the fins inclination is lower. The variable α also has a lower value, which

means the connection line between the fin and the lower plate is smaller and, consequently, the panel

height increases. This can almost be seen as the optimal configuration trying to compensate for the

height lost with the fin inclination. Despite this compensation the height is about 20% lower than

in cases 5.2 and 5.3.

The picture below compares the optimal configurations obtained in 5.2.1, 5.3.1 and the

present case. The first two cells are similar. The only difference is that α is slightly higher in 5.2.1, so

the connection length is bigger. The third cell is the lower from the three.

In the present case, due to the boundary conditions, the plate is more vulnerable to

deformation because its stiffness is significantly

Table 7: Case 4.1 results

Figure 9: (i) Case 2.1 configuration (ii) Case 3.1 configuration (iii) Case 4.1 configuration

9

(i)

(ii)

(iii)

reduced. So the optimal plate configuration has to assume more severe results to increase its

stiffness. This is essentially obtained through the fins inclination because, as in the previous case, it

is the variable that affects the plate behavior the most.

5.4.2. Pressure influence (α, θ, b)

In all the previous cases the optimization process

was performed with a constant pressure. This means that we don’t know if the optimal

configuration depends more on the boundary conditions or on the plate pressure. The tests

above were repeated, with the same boundary conditions, but with different pressures. The results

can be seen below:

Alg. Press.[MPa]

Plastic Strain

bopt [m]

Ɵopt [º] αopt

Fun. Evaluat.

Genetic 3.00 22.83 0.081 58.96 0.268 1040

fmincon 3.00 22.80 0.080 59.04 0.267 178

Genetic 3.25 30.79 0.080 58.83 0.268 890

fmincon 3.25 30.78 0.080 58.99 0.265 201

Genetic 3.50 40.21 0.081 62.10 0.272 1120

fmincon 3.50 39.18 0.081 61.02 0.266 245

The obvious conclusion is that the boundary

conditions control the optimal solution. Despite some slight variations, all the results are essentially

the same, but the plate height is approximately 10% lower in this case than in 5.3.

The only variation is in the objective function. This

is expected because higher pressures cause higher plastic deformation, i.e., higher equivalent plastic

strain.

5.4.3. Linear analysis: Elastic energy (α, θ, b)

A linear static analysis is performed, using these boundary conditions. The objective is to see how

Table 8: Case 4.2 results

the optimal configuration changes. The results are shown below:

Alg. Elastic Energy

bopt [m]

Ɵopt [º] αopt

Fun. Evaluat.

Genetic 0.129 0.088 69.53 0.278 720

fmincon 0.130 0.088 69.30 0.278 184

Comparing these results with 5.1.2 we see that θ is

lower and α is higher, lowering the panel height in about 10%. As mentioned before, these boundary

conditions make the plate more vulnerable to deformation, reducing its stiffness. So θ and α have

to compensate for this lost.

If we compare this solution with 5.4.1 we see that the fins present a lower inclination and α is slightly

lower. Since the pressure applied for the plate to present plastic deformation is significantly higher

than the pressure applied in this case it’s expected that the cells in 5.4.1 have to adopt a more severe

configuration, so it enhances the plate stiffness. This comparison can be seen in the picture below:

5. Conclusion

The main objectives of this work were obtained. A

computational model that allows the plate analysis was ob ta ined, as were severa l op t ima l

configurations for different situations.

The obtained results suggest that θ is the variable which affects the plate properties the most, namely

its stiffness. Cases 5.2, 5.3 and 5.4 show that whenever the boundary conditions cause the plate

to loose its stiffness this is compensated with a

Table 9: Case 4.3 results

Figure 10: (i) Case 4.1 configuration (ii) Case 4.2 configuration

10

(i) (ii)

lower θ. This is also achieved with α, but its influence isn’t so noticeable.

The results also suggest that it’s not desirable for

the plate to have a low height. This can be seen in the cases where, because of the boundary

conditions, the fins present higher θ. In these cases α has a lower value, so that the height loss isn’t so

severe.

The results also suggest that the plate optimal configuration depends, essentially, on the boundary

conditions. They also suggest that, when in the plastic regime, the variation of the applied pressure

doesn’t change the optimal configuration, but when we change the boundary conditions we can have

severe changes. This is an advantage because when a plate is used for the same application,

regardless of the applied pressure, it always has the optimal configuration. With the same boundary

conditions the optimal configuration changes from the elastic to the plastic regime.

Regarding the algorithms used, the results

obtained were very satisfying. The main disadvantage comes with the genetic algorithm

used with a non-linear constraint. As showed, the number of functions evaluations is very high and

the process is very slow. In some cases this may present as a problem.

As future work, these tests can be made with a

different cell geometry, to see how the plate stiffness changes. The plate used in this work is

regular repetition of the same cell, having the same configuration. It would be interesting to perform an

optimization analysis that allows cells from the same plate to have different optimal configurations

between them. Finally, it’s suggested to make the same tests made in this paper but using different

loads, namely shear stresses.

6. References

[1] Luongo, Fabio - “Optimization of the

Reinforcement Configuration of an OPENCELL® Type Flat Panel” - Instituto Superior Técnico, 2011

[2] Love, A. E. H. - On the Small Free Vibrations and Deformations of Elastic Shells - Philosophical

trans. of the Royal Society (London), Vol. série A, N° 17, 1888

[3] Mota Soares, Carlos - “Teoria e Análise de Placas: Métodos Analíticos e Aproximados”

Mecânica Estrutural, 1982

[4] Reissner, E. - “The Effect of Transverse Shear

Deformation on the Bending of Elastic Plates” ASME Journal of Applied Mechanics, Vol. 12, 1945

[5] Rechenberg, Ingo - “Evolutionsstrategie - Optimierung Technischer Systeme Nach Prinzipien

der Biologischen Evolution” Fromman-Holzboog, 1971

[6] Henry Holland, John - “Adaptation in Natural and Artificial Systems”, 1975

[7] S. Arora, Jasbir - Introduction to Optimal Design. 2ª Edição: Elsevier Academic Press, 2004

[8] Dunn, Fionn e Petrinic, Nik - Introduction to Computational Plasticity. 1ª Edição: Oxford

University Press, 2005

[9] von Mises, R. - “Mechanic der Festen Körper in

Plastisch Deformablem Zustand”, Göttinger Nachr. Math. Phys. Kl., 1913

[10] Drucker, D.C. and Prager, W. - “Soil mechanics and plastic analysis or limit design”, Q. J. Appl.

Math., Vol.10, 1952

[11] “Release 11.0 Documentation” Ansys

[12] “Matlab Product R2011a Documentation” Mathworks

11